Calculator Inputs
Example Data Table
| Distribution | Parameters | Threshold | Approx. left tail probability | Reading |
|---|---|---|---|---|
| Standard Normal | μ = 0, σ = 1 | -1.25 | 0.1056 | A low z-score leaves about 10.56% to the left. |
| Normal | μ = 100, σ = 15 | 120 | 0.9088 | The threshold sits around the 90.88th percentile. |
| Student's t | df = 10 | -1.812 | 0.0500 | This threshold is near the lower 5% tail. |
| Chi-square | df = 8 | 6 | 0.3528 | About 35.28% of values fall at or below 6. |
| Binomial | n = 20, p = 0.35 | 5 | 0.4164 | This is the probability of five or fewer successes. |
| Poisson | λ = 4.5 | 3 | 0.3423 | About 34.23% of counts are three or less. |
Formula Used
Normal and Standard Normal
Use P(X ≤ x) = Φ((x - μ) / σ). The standard normal case is the special version with
μ = 0 and σ = 1.
Student's t
The left tail is the cumulative probability from negative infinity to x, computed from the
t-distribution with the chosen degrees of freedom.
Chi-square
Use the regularized gamma form P(X ≤ x) = γ(k/2, x/2) / Γ(k/2), where k is the
degrees of freedom.
Exponential
Use P(X ≤ x) = 1 - e-λx for x ≥ 0. It measures the waiting-time chance
up to the selected threshold.
Binomial
Use P(X ≤ k) = Σ C(n,i) pi(1-p)n-i from i = 0 to
k, where k = floor(x).
Poisson
Use P(X ≤ k) = Σ e-λ λi / i! from i = 0 to
k, where k = floor(x).
How to Use This Calculator
1. Choose the distribution
Select the probability model that matches your data, such as normal, t, chi-square, binomial, or Poisson.
2. Enter the threshold
Provide the value x where you want the left tail probability P(X ≤ x).
3. Add parameters
Fill in the required mean, standard deviation, rate, degrees of freedom, trials, or success probability for the chosen model.
4. Set precision and alpha
Adjust decimal places for reporting and use the significance level to judge whether the threshold lies inside the lower tail.
5. Review outputs and graph
After calculation, read the left tail probability, right tail, percentile rank, interpretation, and the interactive Plotly cumulative curve.
6. Export the result
Use the CSV or PDF buttons to save the result summary for reporting, teaching, or audit records.
Frequently Asked Questions
1. What does left tail probability mean?
It is the cumulative probability that a random variable is less than or equal to a chosen threshold. In symbols, it is P(X ≤ x).
2. Why does the calculator ask for different parameters?
Each distribution has its own shape controls. Normal uses mean and standard deviation, t and chi-square use degrees of freedom, and count models need trial or rate settings.
3. Why are discrete thresholds rounded down?
Discrete models only allow whole counts. A threshold like 5.8 means the same left tail event as 5, so the calculator uses floor(x).
4. When should I use the standard normal option?
Use it when your variable is already expressed as a z-score or when the original normal variable has been standardized with mean zero and standard deviation one.
5. What does the percentile rank show?
Percentile rank converts the left tail probability into a percentage. A 90th percentile means the threshold is greater than about 90% of values under that model.
6. How should I interpret alpha here?
Alpha is used as a comparison guide. If the left tail probability is smaller than alpha, the threshold lies inside the lower alpha tail of the distribution.
7. Does the Plotly graph show probability density?
This graph shows the cumulative distribution function. That makes the left tail easy to read directly because the y-value is the cumulative probability at the selected threshold.
8. Can I use this for classroom and reporting work?
Yes. The page is suitable for study, quick analysis, and reporting. Export tools help capture the result summary in reusable formats.